Stochastic optimal scheduling of a combined wind-photovoltaic-CSP-fire system accounting for electrical heat conversion

Abstract

Aiming at the consumption problem caused by the increasing scale of wind power and photovoltaic (PV) grid-connected, a multi-energy co-generation system is constructed with wind power, PV, concentrated solar power (CSP), and thermal power; in addition, in order to reduce the impact of the prediction errors of wind power, PV, and loads on the system’s economic operation, the photovoltaic and thermal power plants are used to provide the system's backup capacity together, and the opportunity constraint model of the reliability of the backup capacity is established, so as to satisfy the system reliability constraints at a certain higher confidence level; finally, a sampling-based deterministic transformation method is introduced to simplify the model. The model is simplified by introducing a sampling-based deterministic transformation method of opportunity constraint; finally, a stochastic optimal dispatch model of the combined wind-photovoltaic-CSP-fire system, which accounts for the conversion of electricity and heat, is constructed with the objective of minimising the integrated operating cost of the combined system, and the effectiveness of the proposed model is analysed by simulation verification.

Introduction

With the increasing penetration of wind power generation, as of the end of 2019, China's wind power grid total exceeded 2.1 × 10⁸ kW (Baharoon et al. 2015), the total installed capacity of wind power is 2.36 × 10⁸ kW, however, the wind power does not have a water, thermal power generation of the stability and man-made controllability leads to its grid to the system causes great trouble (Pandzic et al. 2016; Zhang et al. 2021). Renewable energy uncertainty in the system scheduling specifically in the prediction error, the current wind power, photovoltaic and load forecasting research has reached a high level, but can not completely eliminate the error, the need for the power system to set aside sufficient rotating standby capacity in order to cope with the prediction error on the operation of the system brought about by the impact of the system, in addition to the need to take into account the fluctuation of the wind power, photovoltaic power (Xiang et al. 2021a), load fluctuations, as well as the failure of the unit and other uncertainties, so that in the new energy-containing combined system of the standby capacity is particularly important to the analysis of the standby capacity (Xu et al. 2021; Elghitani and Zhuang 2018).

Literature (Balasubramanian and Balachandra 2021) proposed an optimal scheduling strategy to maximize the economic benefits of the wind-light-fire-storage combined system, which comprehensively considered the cost of wind and light abandonment and the difficulty of peak load balancing of thermal power units, and ensured the economic operation of the multi-source combined system through the optimal energy abandonment rate. Literature (Xu and Ning 2016) proposed a wind-wind-fire combined dispatch method from the perspective of economic and environmental protection balance, which fully considered the environmental benefits brought by grid-connected wind power and photovoltaic power generation, effectively improved the consumption level of wind power and photovoltaic power generation, and reduced the coal consumption rate. Literature (Liu et al. 2019) proposes a wind-wind-storage joint optimal scheduling model considering load distribution strategy, which maximally matches the output of wind power and photovoltaic power generation with the electricity demand, and improves the utilization rate of wind power and photovoltaic power generation and the economy of system operation. Literature (Yang et al. 2021) proposes a collaborative peaking operation strategy of wind-light-cascade hydropower station, which can improve the system's absorption rate of wind power and photovoltaic power generation by coordinating the output of cascade hydropower station. Literature (Xiang et al. 2021b) established an optimized scheduling model of wind-light-water-fire-pumping and storage combined power generation system, which comprehensively considered the cost of wind and light abandonment and environmental benefits. Through the coordination and cooperation of various energy sources, the phenomenon of wind and light abandonment was reduced, and the comprehensive operation cost of the system was effectively reduced. The above literature studies have proved the effectiveness of thermal power generation, energy storage devices and pumped storage power stations in improving the consumption level of wind power and photovoltaic, but the conventional thermal power generation is mostly coal-fired units, which is not conducive to protecting the environment and reducing carbon dioxide emissions. The response speed of battery energy storage is fast, but the cost is too high, and pumped storage power stations are restricted by geographical environment.

In this paper, uncertainty variables such as forecast errors of wind power, photovoltaic and load are considered, and the current methods of dealing with uncertainty variables mainly include interval optimization (Liu et al. 2019; Kirkerud et al. 2021; Chen et al. 2015), robust optimisation and stochastic optimisation. The interval optimisation method mainly characterises the distribution of uncertain variables in terms of the number of intervals to be involved in the optimisation decision, while the robust optimisation method mainly represents the uncertain variables in terms of a certain set of variation ranges and formulates the decision in terms of the worst-case Scene, and both the interval optimisation method and the robust optimisation method are generally more conservative in the formulation of the decision, which results in poorer economy (Nadeem et al. xxxx). The stochastic optimisation method is mainly based on the probability distribution function of the uncertainty variables to optimise, which allows some of the decisions to not fully satisfy the constraints, but the probability of its establishment must meet a certain confidence level, and through the reasonable setting of the confidence level can effectively improve the economy of the system operation under the premise of ensuring system safety (Zz et al. 2020).

Standby capacity modelling for joint systems based on opportunity constrained planning

Opportunity constrained planning

Opportunity constraint programming, as a stochastic optimisation method (Gharibi et al. 2019), can allow some of the decisions to not fully satisfy the constraints, but the probability that they hold must satisfy a certain confidence level, and the standard form of opportunity constraint programming is:

$$\left\{ \begin{gathered} \max f(x,\varepsilon ) \hfill \\ s.t. \hfill \\ P_{r} \left\{ {g_{i} (x,\xi ) \le 0(j = 1,2,...,p)} \right\} \ge \alpha \hfill \\ G_{i} (x,\xi ) \le 0 \hfill \\ \end{gathered} \right.$$

(1)

where $f(x,{\upvarepsilon })$is the objective function; x is the decision vector;$\varepsilon$ is the stochastic vector; ${\text{g}}_{{\text{i}}} \left( {{\text{x,}}\xi } \right)$is the chance constraint function${\text{P}}_{r} \{ g_{i} (x,\zeta ) \le 0(j = 1,2, \cdots ,P)\}$ is the probability of the event being established; for α is the confidence level;${\text{G}}_{{\text{i}}} \left( {{\text{x,}}\xi } \right)$is the rigidity constraint.

When using opportunity constrained planning theory to deal with the system rotating spare capacity problem (Lei et al. 2017), it can be expressed in the following form:

$$P_{r} \{ P_{{{\text{R,}}t}}^{{}} + \varepsilon_{t} \ge P_{{{\text{Req}},t}}^{{}} \} \ge \beta_{{}}$$

(2)

where ${\text{P}}_{R,t}$is the reserved reserve of the system at moment t;$\varepsilon_{t}$ is the forecast error at moment $t;P_{Raq,t}$ is the reserve demand at moment t, and β is the confidence level.

Forecast errors

$$\left\{ \begin{gathered} P_{{\text{t}}}^{{\text{W}}} = P_{{\text{t}}}^{{{\text{WP}}}} + \Delta P_{t}^{{\text{W}}} \hfill \\ P_{{\text{t}}}^{{{\text{PV}}}} = P_{{\text{t}}}^{{{\text{PVP}}}} + \Delta P_{t}^{{{\text{PV}}}} \hfill \\ P_{{\text{t}}}^{{\text{L}}} = P_{{\text{t}}}^{{{\text{LP}}}} + \Delta P_{t}^{{\text{L}}} \hfill \\ \end{gathered} \right.$$

(3)

At present, the empirical distribution of the prediction error of wind power generation mainly includes normal distribution, Beta distribution, Cauchy distribution. Since the wind power prediction error is affected by many types of factors such as geography and climate, for the large-scale wind farm cluster with wide geographical distribution, its overall power prediction error is mostly described by normal distribution, and the PV prediction error as well as the load prediction error are also mostly described by normal distribution. Therefore, in this paper, normal distribution is used to represent the prediction errors of wind power, PV and load, and the mean and standard deviation of the three are:$^{\mu } WT\,^{\mu } PV^{\mu } Load$ and$^{\sigma } WT^{\,\sigma } PV$ respectively, $\sigma_{Load}$ the prediction error of wind power output $\begin{gathered} \Delta P_{t}^{WT} \hfill \\ \Delta P_{t}^{WT} \hfill \\ \end{gathered}$ at moment t can be written as$X_{WT} \sim (^{\mu } WT\,\sigma_{WT}^{2} \,)$ the prediction error of PV power output $\Delta P_{t}^{PV}$at moment t can be written as $X_{PV} \sim (\mu_{PV} ,\sigma_{PV}^{2} )$ the load forecast error $\Delta P_{t}^{L}$ at moment t can be written as $X_{Load} \sim^{{\left( {{}^{\mu }Load_{Load}^{\sigma 2} } \right)}}$:

$$\left\{ \begin{gathered} X_{{{\text{WT}}}} \sim N(\mu_{{{\text{WT}}}} ,\sigma_{{{\text{WT}}}}^{2} ) \hfill \\ X_{{{\text{PV}}}} \sim N(\mu_{{{\text{PV}}}} ,\sigma_{{{\text{PV}}}}^{2} ) \hfill \\ X_{{{\text{Load}}}} \sim N(\mu_{{{\text{Load}}}} ,\sigma_{{{\text{Load}}}}^{2} ) \hfill \\ \end{gathered} \right.$$

(4)

$$\left\{ \begin{gathered} f(X_{{{\text{WT}}}} ) = \frac{1}{{\sqrt {2\pi } \sigma_{{{\text{WT}}}} }}e^{{ - \frac{{(X_{{{\text{WT}}}} - \mu_{{{\text{WT}}}} )^{2} }}{{2(\sigma_{{{\text{WT}}}} )^{2} }}}} \hfill \\ f(X_{{{\text{PV}}}} ) = \frac{1}{{\sqrt {2\pi } \sigma_{{{\text{PV}}}} }}e^{{ - \frac{{(X_{{{\text{PV}}}} - \mu_{{{\text{PV}}}} )^{2} }}{{2(\sigma_{{{\text{PV}}}} )^{2} }}}} \hfill \\ f(X_{{{\text{Load}}}} ) = \frac{1}{{\sqrt {2\pi } \sigma_{{{\text{Load}}}} }}e^{{ - \frac{{(X_{{{\text{Load}}}} - \mu_{{{\text{Load}}}} )^{2} }}{{2(\sigma_{{{\text{Load}}}} )^{2} }}}} \hfill \\ \end{gathered} \right.$$

(5)

where, $\mu_{WT} ,\,\mu_{PV} ,\,\mu_{Load}$ are the mean values of wind, PV, and load, respectively; $\sigma_{WT} ,\sigma_{PV} ,\,\sigma_{Load} \,$are the standard deviations of wind, PV, and load, respectively.

Spare capacity modelling

Solar thermal power plants and thermal power units are used to provide system standby capacity together (Zz et al. 2020), and a joint system standby reserve capacity model is constructed based on opportunity constrained planning as shown in Eq. (6) and Eq. (7), and the standby capacity is optimised so that it can satisfy the reliability model at a certain higher confidence level.

$$P_{r} \{ P_{{{\text{R,}}t}}^{{{\text{up}}}} + \sum\limits_{i = 1}^{{N_{{{\text{WT}}}} }} {\Delta P_{i,t}^{{{\text{WT}}}} + \sum\limits_{i = 1}^{{N_{{{\text{PV}}}} }} {\Delta P_{i,t}^{{{\text{PV}}}} } } - \Delta P_{t}^{{\text{L}}} \ge P_{{{\text{Res}},t}}^{{{\text{up}}}} \} \ge \beta_{1}$$

(6)

$$P_{r} \{ P_{{{\text{R,}}t}}^{{{\text{down}}}} - \sum\limits_{i = 1}^{{N_{{{\text{WT}}}} }} {\Delta P_{i,t}^{{{\text{WT}}}} - \sum\limits_{i = 1}^{{N_{{{\text{PV}}}} }} {\Delta P_{i,t}^{{{\text{PV}}}} } } + \Delta P_{t}^{{\text{L}}} \ge P_{{{\text{Res}},t}}^{{{\text{down}}}} \} \ge \beta_{2}$$

(7)

$$\left\{ \begin{gathered} P_{{{\text{R,}}t}}^{{{\text{up}}}} = P_{{{\text{G}},t}}^{{{\text{up}}}} + P_{{{\text{CSP}},t}}^{{{\text{up}}}} \hfill \\ P_{{{\text{R,}}t}}^{{{\text{down}}}} = P_{{{\text{G}},t}}^{{{\text{down}}}} + P_{{{\text{CSP}},t}}^{{{\text{down}}}} \hfill \\ \end{gathered} \right.$$

(8)

where $P_{R;t,}^{up} \,P_{R;t}^{down}$ are the positive and negative rotating reserve capacity of the system at moment t, respectively;$\Delta P_{i;t}^{WT} ,\,\Delta P_{i;t}^{PV} \,and\,\Delta P_{t}^{L}$are the fluctuating power of wind, PV and load at moment t, respectively; and $P_{{{\text{Re}} s;t}}^{up} \,,\,P_{{{\text{Re}} s;\,t}}^{down}$ are the standby demand at moment t, respectively. $\beta_{1} ,\,\beta_{2}$ are the confidence levels required to be satisfied by the positive and negative rotating reserve capacity opportunity constraint, respectively.

The traditional stochastic simulation approach to solving opportunity constraint planning is computationally intensive and inefficient, so this paper adopts a sampling-based deterministic transformation method of chance constraints for the transformation.

Firstly, the scenes are generated according to the probability distributions obeyed by each of the random variables$\Delta P_{i,t}^{WT} ,\,\Delta P_{i,t}^{PV} \,and\,\Delta P_{t}^{L}$ and they are discretised with Latin hypercubic sampling. Let the sampling number be N, then the sth sampling value is recorded as$\Delta P_{S,i,t,\,\,}^{WT} \Delta P_{s,i,t}^{PV} \,and\,\Delta P_{s,t}^{L}$ When the sampling number is large enough, then Eq. (6) and (7) can be transformed into the mixed integers shown in Es. (9) to (12) Linear constraints.

$$P_{{{\text{R,}}t}}^{{{\text{up}}}} + \sum\limits_{i = 1}^{{N_{{{\text{WT}}}} }} {\Delta P_{s,i,t}^{{{\text{WT}}}} + \sum\limits_{i = 1}^{{N_{{{\text{PV}}}} }} {\Delta P_{s,i,t}^{{{\text{PV}}}} } } - \Delta P_{s,t}^{{\text{L}}} \ge M(1 - d_{1,t} (s))$$

(9)

$$P_{{{\text{R}},t}}^{{{\text{down}}}} - \sum\limits_{i = 1}^{{N_{{{\text{WT}}}} }} {\Delta P_{s,i,t}^{{{\text{WT}}}} - \sum\limits_{i = 1}^{{N_{{{\text{PV}}}} }} {\Delta P_{s,i,t}^{{{\text{PV}}}} } } + \Delta P_{s,t}^{{\text{L}}} \ge M(1 - d_{2,t} (s))$$

(10)

$$\sum\limits_{s = 1}^{N} {d_{1,t} (s)} \ge N\beta_{1}$$

(11)

$$\sum\limits_{s = 1}^{N} {d_{2,t} (s)} \ge N\beta_{2}$$

(12)

where M is a sufficiently small negative number. If$d_{1} ,t(s)$ is 1, the reserve capacity inequality constraint in Eq. (11) is valid, and if 0 is not valid;$d_{2} ,t(s)$ is 1, the reserve capacity inequality constraint in Eq. (12) is valid, and 0 is not valid.

The optimisation variable and the random variable in Eqs. (9) and (10) can be separated, so that Eqs. (9) and (10) can be further simplified by combining and then sorting the sampled values of the random variables to obtain the following short form.

$$P_{{{\text{R,}}t}}^{{{\text{up}}}} + \Delta P_{t} ({\text{floor(}}N(1 - \beta_{1} ){)}) \ge 0$$

(13)

$$P_{{{\text{R,}}t}}^{{{\text{down}}}} - \Delta P_{t} ({\text{ceil(}}N(\beta_{2} ){)}) \ge 0$$

(14)

$$\Delta P_{t} = {\text{sort([}}\Delta P_{t}^{1} \, \Delta P_{t}^{2} \, \Delta P_{t}^{3} \cdot \cdot \cdot \Delta P_{t}^{s} \cdot \cdot \cdot \Delta P_{t}^{N} {])}$$

(15)

$$\Delta P_{t}^{s} = \sum\limits_{i = 1}^{{N_{{{\text{WT}}}} }} {\Delta P_{s,i,t}^{{{\text{WT}}}} + \sum\limits_{i = 1}^{{N_{{{\text{PV}}}} }} {\Delta P_{s,i,t}^{{{\text{PV}}}} } } - \Delta P_{s,t}^{{\text{L}}}$$

(16)

where floor (*) is the downward rounding function; ceil(*) is the upward rounding function; sort (*) is the ascending sorting function; $\Delta P, \cdots ,\Delta P_{t}^{N}$ denote the net load power of the Scene s(s = 1, 2, ⋅⋅⋅, N) at the moment of t, while $\Delta P_{t}$ denotes the vector formed by arranging the net load power under N Scenes corresponding to the moment of t according to the ascending sorting.

Stochastic optimal scheduling model for a combined wind-photovoltaic-CSP-fire system taking into account the conversion of electricity and heat

Objective function

Establishes an optimal operation model for wind-photovoltaic-CSP-fire co-generation system with the optimisation objective of the lowest comprehensive operating cost of the system. The objective function mainly consists of the following cost components:

$$\min f = C_{1} + C_{2} + C_{3} + C_{4} + C_{5} + C_{6} + C_{7}$$

(17)

${C}_{1}$ is the cost of generating electricity from thermal power units:

$$C_{1} = \sum\limits_{t = 1}^{T} {\sum\limits_{i = 1}^{{N_{{\text{G}}} }} {\left[ {(a_{i} (P_{i,t}^{{\text{G}}} )^{2} + b_{i} P_{i,t}^{{\text{G}}} + c_{i} ) + } \right.} } \left. {u_{i,t} (1 - u_{i,(t - 1)} )S_{i} } \right]$$

(18)

where ${a}_{i}, {b}_{i}$ and ${c}_{i}$ are the fuel cost coefficients of the thermal power units;$P_{i,\,t}^{G}$ is the output power of the ith thermal power unit at time t; ${u}_{i,t}$ is the start-stop state of the thermal power unit, and $u_{i,t}$ is 0 indicating that the thermal power unit is shut down; and ${S}_{i}$ is the start-stop cost of the ith thermal power unit.

${C}_{2}$ is the cost of wind farm operation and maintenance:

$$C_{2} = \sum\limits_{t = 1}^{T} {\sum\limits_{i = 1}^{{N_{{{\text{WT}}}} }} {\gamma_{{{\text{WT}}}} P_{i,t}^{{{\text{WT}}}} } }$$

(19)

where $\gamma_{WT}$ is the unit operation and maintenance cost of the wind power system;$P_{i,t}^{WT}$ is the output power of the ith wind turbine at the moment t.

${\text{C}}_{3}$ is the cost of PV plant operation and maintenance:

$$C_{3} = \sum\limits_{t = 1}^{T} {\sum\limits_{i = 1}^{{N_{{{\text{PV}}}} }} {\gamma_{{{\text{PV}}}} P_{i,t}^{{{\text{PV}}}} } }$$

(20)

where $\gamma_{PV}$ is the unit operation and maintenance cost of the PV system;$P_{i,t}^{PV}$ is the output power of the ith group of PV arrays at time t.

${\text{C}}_{4}$ is the cost of operation and maintenance of the photovoltaic power system:

$$C_{4} = \sum\limits_{t = 1}^{T} {\gamma_{{{\text{CSP}}}} P_{t}^{{{\text{CSP}}}} }$$

(21)

where $\gamma_{{{\text{CSP}}}}$ is the unit operation and maintenance cost of the photovoltaic power plant;$P_{t}^{CSP}$ is the output power of the photovoltaic power generation system at moment t.

${\text{C}}_{5}$ is the cost of rotating the standby for the system:

$$\begin{gathered} C_{5} = \sum\limits_{t = 1}^{T} {\sum\limits_{i = 1}^{{N_{{\text{G}}} }} {(k_{{\text{G,up}}} P_{{{\text{G}},i,t}}^{{{\text{up}}}} + k_{{\text{G,down}}} P_{{{\text{G}},i,t}}^{{{\text{down}}}} ) + } } \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} \sum\limits_{t = 1}^{T} {(k_{{\text{CSP,up}}} P_{{{\text{CSP}},t}}^{{{\text{up}}}} + k_{{\text{CSP,down}}} P_{{{\text{CSP}},t}}^{{{\text{down}}}} )} \hfill \\ \end{gathered}$$

(22)

where $K_{G,\,up} ,\,K_{G,down}$ are the cost coefficients of positive and negative rotating reserve capacity reserved for thermal power units, respectively; $K_{CSP,up} ,\,K_{CSP,\,down}$ are the cost systems of positive and negative rotating reserve capacity reserved for photovoltaic power generation systems, respectively;$P_{G,i,t}^{up} ,P_{G,i,t}^{down}$ are the positive and negative rotating reserve capacity reserved for thermal power units, respectively; $P_{CSP,t}^{up} \,and\,P_{CSP,t}^{down}$ are the positive and negative rotating reserve capacity reserved for photovoltaic power generation systems, respectively.

${\text{C}}_{6}$ is the Wind and Light Penalty Charge:

$$\begin{gathered} C_{6} = \sum\limits_{t = 1}^{T} {\sum\limits_{i = 1}^{{N_{{{\text{PV}}}} }} {\delta_{{\text{pv,w}}} (P_{i,t}^{{\text{P,E}}} - P_{i,t}^{{{\text{PV}}}} - P_{i,t}^{{\text{P,P - H}}} ) + } } \, \hfill \\ \, \sum\limits_{t = 1}^{T} {\sum\limits_{i = 1}^{{N_{{{\text{WT}}}} }} {\delta_{{\text{wt,w}}} (P_{i,t}^{{\text{W,E}}} - P_{i,t}^{{{\text{WT}}}} - P_{i,t}^{{\text{W,W - H}}} )} } \hfill \\ \end{gathered}$$

(23)

where$\delta_{Wt,w} ,\delta_{pv,w}$ are the penalty factors for wind and light abandonment, respectively;$P_{i,t}^{P,E} ,\,P_{E,i,t}^{W}$ are the output power of the ith device of the PV power system and the wind power system, respectively; $P_{i,t}^{PV} ,\,P_{i,t}^{WT}$ are the grid-connected power of the PV and the wind power, respectively$P_{i,t}^{P,P - H}$ and$P_{i,t}^{W,W - H}$ are the power of the wind system and the PV system feeding into the electric heating at the moment t, respectively.

${\text{C}}_{7}$ is the cost of electric heating:

$$C_{7} = K_{EH} (P_{i,t}^{{\text{P,P - H}}} + P_{i,t}^{{\text{W,W - H}}} )$$

(24)

where ${\text{K}}_{\text{EH}}$ is the cost coefficient of electric heat conversion.

Restrictive condition

(1) Power balance constraint:

$$\sum\limits_{{i{ = 1}}}^{{N_{{{\text{WT}}}} }} {P_{i,t}^{{{\text{WT}}}} } + \sum\limits_{i = 1}^{{N_{{{\text{PV}}}} }} {P_{i,t}^{{{\text{PV}}}} } + \sum\limits_{i = 1}^{{N_{{{\text{CSP}}}} }} {P_{i,t}^{{{\text{CSP}}}} } + \sum\limits_{i = 1}^{{N_{{\text{G}}} }} {P_{i,t}^{{\text{G}}} } = P_{t}^{{\text{L}}}$$

(25)

where $P_{i,t}^{WT} ,\,P_{i,t}^{PV} ,\,P_{i,t}^{CSP} ,\,P_{i,t}^{G}$ are the output power of the ith wind turbine, photovoltaic generator set, photothermal generator set, and thermal generator set at time t, respectively; and ${\text{P}}_{\text{t}}^{\text{L}}$ is the actual load power at time t.

(2) Thermal unit constraints:

$$\left\{ \begin{gathered} x_{i,t}^{{\text{G}}} P_{i,\min }^{{\text{G}}} \le P_{i,t}^{{\text{G}}} \le x_{i,t}^{{\text{G}}} P_{i,\max }^{{\text{G}}} \hfill \\ P_{i,t}^{{\text{G}}} - P_{i,t - 1}^{{\text{G}}} \le R_{{\text{d}}}^{{\text{G}}} \hfill \\ \end{gathered} \right.$$

(26)

where $x_{i,t}^{G}$ is the operating state of the thermal power unit, and a value of 1 indicates that the generating unit starts, and a value of 0 indicates that the generating unit stops; $P_{i,\max }^{G} ,\,P_{i,Min}^{G}$ are the upper and lower limits of the output of the ith thermal power unit, respectively; and$R_{d}^{G}$ is the creepage rate of the thermal power unit.

(3) Wind power constraints:

$$P_{\min }^{{{\text{WT}}}} \le P_{i,t}^{{{\text{WT}}}} \le P_{\max }^{{{\text{WT}}}}$$

(27)

where $P_{\min }^{WT} \,and\,P_{\max }^{WT}$ PWT min and PWT max are the upper and lower limits of the output power of the wind farm, respectively.

(4) Power constraints for PV power systems:

$$P_{\min }^{{{\text{PV}}}} \le P_{i,t}^{{{\text{PV}}}} \le P_{\max }^{{{\text{PV}}}}$$

(28)

where $P_{\min }^{PV} \,and\,P_{\max }^{PV}$ are the upper and lower limits of the output power of the PV plant, respectively.

(5) The system rotating standby constraints are shown in the previous Eqs. (6) to (8).

(6) Thermal storage system constraints:

$$\left\{ \begin{gathered} 0 \le Q_{{{\text{c}},t}}^{{\text{TES,ch}}} \le Q_{{\text{c}}}^{{\text{TES,max}}} x_{{{\text{c}},t}}^{{{\text{ch}}}} \hfill \\ 0 \le Q_{{{\text{c}},t}}^{{\text{TES,dch}}} \le Q_{{\text{c}}}^{{\text{TES,max}}} x_{{{\text{c}},t}}^{{{\text{dch}}}} \hfill \\ x_{{{\text{c}},t}}^{{{\text{ch}}}} + x_{{{\text{c}},t}}^{{{\text{dch}}}} \le 1 \hfill \\ x_{{{\text{c}},t}}^{{{\text{ch}}}} ,x_{{{\text{c}},t}}^{{{\text{dch}}}} \in \{ 0,1\} \hfill \\ \end{gathered} \right.$$

(29)

In the formula,$Q_{c}^{TES,\max }$ is the upper limit of heat storage power of the heat storage system;$x_{c,t}^{ch} ,\,x_{c,t}^{dch}$ are the heat storage and exothermic state respectively, and the heat storage system can only work in the heat storage or exothermic state at the same moment.

(7) CSP generation unit constraints:

$$\left\{ \begin{gathered} x_{t}^{{{\text{PC}}}} P_{t,\min }^{{{\text{PC}}}} \le P_{t}^{{{\text{PC}}}} \le x_{t}^{{{\text{PC}}}} P_{t,\max }^{{{\text{PC}}}} \hfill \\ P_{t}^{{{\text{PC}}}} - P_{t - 1}^{{{\text{PC}}}} \le (1 - u_{t}^{{{\text{PC}}}} )R_{{\text{U}}}^{{{\text{PC}}}} + u_{t}^{{{\text{PC}}}} P_{t,\min }^{{{\text{PC}}}} \hfill \\ P_{t - 1}^{{{\text{PC}}}} - P_{t}^{{{\text{PC}}}} \le (1 - v_{t}^{{{\text{PC}}}} )R_{{\text{U}}}^{{{\text{PC}}}} + v_{t}^{{{\text{PC}}}} P_{t,\min }^{{{\text{PC}}}} \hfill \\ u_{t}^{{{\text{PC}}}} + v_{t}^{{{\text{PC}}}} \le 1 \hfill \\ \end{gathered} \right.$$

(30)

where, ${\text{x}}_{\text{t}}^{\text{PC}}$ is the operation state of the photovoltaic power generation unit, and 1 indicates that the generating unit starts, and 0 indicates that the generating unit stops;$P_{t,\max }^{PC} ,\,P_{t,\min }^{PC}$ are the upper and lower limits of the power of the gas turbine, respectively; $u_{t}^{PC} ,\,v_{t}^{PC}$ denote the startup and stopping state of the gas turbine, respectively; and $R_{U}^{PC}$ denotes the rate of climb of the gas turbine.

(8) Maximum power constraints on transmission lines

$${ - }P_{{\text{l,max}}} \le P_{{{\text{grid,}}t}} \le P_{{\text{l,max}}}$$

(31)

where $P_{grid,t} ,\,P_{l,\max }$ are the actual and maximum power of the contact line at time t, respectively.

Model solution

In this paper, the YALMIP toolkit is used for modelling and the CPLEX solver is invoked for solving. The solution process is shown in Fig. 1, and the specific steps are described below:

Fig. 1

Model solving flow

Step 1: Generate wind, PV and load error Scenes according to the method described in “, and perform Latin hypercubic sampling on the error Scenes, merge the error sequences under each Scene to obtain ΔPs t and finally sort the Ps t under N Scenes;

Step 2: Substitute Eq. (13) and Eq. (14) into YALMIP to construct the standby capacity optimisation model;

Step 3: Substituting wind power, PV and load forecast data, standby capacity, and system operation related parameters into the optimisation model;

Step 4: Construct the set of optimisation variable constraints according to the constraints in the paper, and construct the optimisation objective function of the wind-photovoltaic-CSP-fire co-generation system according to Eq. (17);

Step 5: Call the CPLEX solver to solve the model and output the optimisation results.

Analysis of examples

Basic data

This paper takes the IEEE 30-node system as an example for simulation analysis, as shown in Fig. 2, in which No.1 thermal power is replaced by a wind farm, No.2 thermal power is replaced by a wind farm, and No.3 thermal power is replaced by a photovoltaic power plant; the installed capacity of the wind farm in the calculation example is 150 MW; the installed capacity of the PV power plant is 150 MW; with reference to the literature, $\gamma_{WT} ,\,\gamma_{PV} ,\,\gamma_{CSP}$ are taken as 20, 30, respectively, 40 RMB/MW; typical day wind power, PV and load forecast curve is shown in Fig. 3, and solar radiation intensity is shown in Fig. 4; the parameters of CSP power plant are shown in Table 1; the parameters of thermal power unit are shown in Table 2; and the values of other parameters are shown in Table 3.

Fig. 2

IEEE 30-node system wiring diagram

Fig. 3

Wind, PV and load forecast curves for a typical day (Balasubramanian and Balachandra 2021; Xu and Ning 2016; Liu et al. 2019)

Fig. 4

Intensity of solar radiation (Yang et al. 2021)

Table 1 CSP power station parameters

Table 2 Parameters of thermal power units

Table 3 Optimization parameters

Analysis of results

(1) Analysis of the results of different deterministic transformation methods.

In order to verify the effectiveness and superiority of the sampling-based deterministic transformation method of opportunity constraint proposed in this paper, three methods are used to solve the problem by comparing and calculating with the traditional analytical method and simulation method under the premise of other conditions being the same.

Method 1: The sampling-based deterministic transformation method of chance constraints proposed in this paper is adopted, based on the characteristics of Monte Carlo algorithm: after exhausting all solutions, the determination of unknown variables can be completed, although a little conservative, but ensure that the solution is established, and the sampling number N is 1000;

Method II: the deterministic transformation of chance constraints is realised using the analytical method, and both Method I and Method II are solved using yalmip calls to Cplex 12.6.3;

Method III: Monte Carlo simulation combined with intelligent algorithm is used to solve the opportunity constraint model, in this paper, the intelligent algorithm uses genetic algorithm, and the parameters of the genetic algorithm are set as shown in Table 4, because the results of genetic algorithm will be different each time the optimisation, so it is taken as the average of the results of 20 times.

Table 4 Genetic algorithm parameters

A comparison of the calculation results of the three different methods is shown in Table 5.

Table 5 Comparison of calculation results

As can be seen from Table 5, the method proposed in this paper has an absolute advantage in the speed of solving the model, while the simulation method has the longest solution time, which is due to the use of genetic algorithm iterative optimisation, each round of the optimisation results have to carry out N times of stochastic simulation, so it greatly increases the model solution time, and the quality of the obtained solution is poor.

Although the solution time and optimisation results of the analytical method are closer to those of the method proposed in this paper, there is still a certain gap, and the analytical method cannot be directly used to transform the chance constraints deterministically when there are multiple independent random variables in the model and each random variable obeys different distributions. This paper uses the sampling-based Scene method to achieve the deterministic transformation of opportunity constraint can effectively break through the traditional analytical method on the number of random variables and the distribution of the law of the limitations, so the method proposed in this paper has obvious advantages over the traditional simulation method and analytical method.

(2) Analysis of optimisation results under different Scenes.

In order to further verify the effectiveness of the stochastic optimisation model proposed in this paper in reducing the amount of wind and light abandonment, the rotating standby cost and the comprehensive operating cost of the system, the following three operating Scenes are set up for comparative analysis.

Scene 1: CSP generation system is not included, the thermal power unit is involved in the standby and the standby mode adopts the traditional way to reserve a fixed standby capacity.

Scene 2: CSP generation system is introduced, but does not include the electric heating device, the CSP system is involved in the standby together with the thermal power unit and the standby model proposed in this paper is adopted.

Scene 3: CSP system and electric heating device are introduced, CSP system and thermal power unit participate in standby together and the standby model proposed in this paper is adopted.

The system cost comparisons under the three Scenes are shown in Table 6.

Table 6 Comparison of system costs under different Scenes

From Table 6, it is known that there are obvious differences in the amount of wind and light abandonment, the cost of rotating standby and the comprehensive operating cost of the system under the three Scenes, with Scene 3 having the lowest comprehensive operating cost, Scene 2 the next highest, and Scene 1 the highest cost. Compared with Scene 1, the cost of Scene 2 is reduced by 1.87%, and the total cost of Scene 3 is reduced by 8.89%.

In order to investigate the reasons that lead to the differences in the integrated cost of the system under the three operating Scenes, the scheduling results of the corresponding economic models for the three Scenes of the system are presented in Fig. 5.

Fig. 5

Scheduling results of different Scenes corresponding to economic models

Combined with Table 6 and Fig. 5, the analysis is as follows:

Comparing the scheduling results of Scene 1 and Scene 2 in Fig. 5, it can be seen that the amount of wind and light abandonment with the participation of CSP power generation system is significantly smaller than the amount of wind and light abandonment without the participation of CSP power generation system because the CSP generator set has a better climbing rate and the CSP power generation system is equipped with heat storage tanks, which has a certain degree of adjustability, and it can level out the fluctuation of wind and PV in a certain time scale, which makes the amount of wind and light abandonment in Scene 2 reduced by 59.7% compared with Scene 1. The amount of wind and light discarded in Scene 2 is reduced by 59.7% compared with Scene 1; comparing the scheduling results of Scene 2 and Scene 3 in Fig. 5, it can be seen that the amount of wind and light discarded is further reduced by the introduction of the electric heating device at the same time as the participation of the CSP system because the electric heating device has the ability of electricity to heat, and part of the wind and light discarded energy can be turned into heat through electric-heat conversion and stored in the storage tank, which reduces the amount of wind and light discarded in Scene 3 compared with that in Scene 1. The amount of wind and light discarded in Scene 3 is reduced by 87.6% compared to Scene 1, and by 69.2% compared to Scene 2.

Comparing the power output of the thermal power units in the three Scene s in the figure, it can be seen that the power fluctuation of the thermal power units with the CSP generation system is significantly smaller than that without the CSP generation system. In the peak load period, the thermal power unit with CSP power generation maintains a smooth output, while the thermal power unit without CSP power generation needs to increase the output power to meet the load demand, and the CSP power generation system plays the role of energy buffer, while the CSP power generation system without CSP power generation system causes the thermal power unit to make frequent adjustments, which increases the additional regulation cost, making the power generation cost of the thermal power unit in Scene 2 reduced by 8.3% compared with Scene 1; in addition, the electric heating device is significantly smaller than that in Scene 1. 8.3%; in addition, the introduction of the electric heating device adds a new heat source to the heat storage system, and the increase in the amount of heat storage improves the reserve capacity of the CSP generation system, which further reduces the power generation cost of the thermal power unit, making the power generation cost of the thermal power unit in Scene 3 10.6% less than Scene 1, and 8.3% less than that of the thermal power unit in Scene 2.

In addition, there is a significant difference in the system spinning standby cost in the three Scenes, with Scene 2 and Scene 3 reduced by 6.3% and 18.9%, respectively, compared to Scene 1. Analysis shows that this is due to the introduction of CSP power generation equipment with stronger reserve capacity and lower operating cost in Scenes 2 and 3, which reduces the standby power borne by thermal power units, and the introduction of electric heating device in Scene 3 increases the energy supply of the heat storage system, which further compresses the proportion of thermal power output, and thus Scene 3 has the lowest standby cost. Meanwhile, the stochastic optimisation model proposed in this paper is used to configure the standby capacity in Scene 2 and Scene 3, which ensures the reasonableness and reliability of the capacity configuration results by simulating the operation under multiple Scenes, and reduces the standby cost under the premise of controlling the operational risk of the system.

In order to further study, the operation of the combined power generation system under the method proposed in this paper, the thermal storage system in scene 3 is analyzed. The change curve of the charging and releasing power and storage capacity of the thermal storage system is shown in Fig. 6.

Fig. 6

Status of CSP thermal storage system

From Fig. 6, it can be seen that from 0:00 to 8:00, the heat storage device provides the heat required for the minimum power operation of the gas turbine. From 9:00 to 16:00, the weather conditions are good and the light radiation is sufficient, and the heat storage device obtains a large amount of heat from the collector and electric heating device and stores it in the hot tank. At 00, this is the late peak of load, and the heat storage system releases heat to assist the thermal power unit in peak regulation. The power fluctuations of wind power and photovoltaic power generation have a great impact on the operation of the unit, while the solar thermal power generation system can smooth the long-term power fluctuations of the system by adjusting the output of the turbine unit and the heat storage and discharge of the heat storage system. At the same time, the heat storage system can keep the same heat storage at the beginning and end of the whole scheduling cycle to prepare for the next scheduling cycle.

Conclusion

Wind power, photovoltaic power, solar thermal power and thermal power form a multi-source co-generation system, with the solar thermal power plant and the thermal power unit together to provide the system spinning reserve capacity and establish a system reserve capacity opportunity constraint model, so that it meets the system reliability constraints at a certain higher confidence level; finally, a wind-photovoltaic-CSP-fire co-generation system optimal scheduling model is constructed based on opportunity constraint planning to The proposed model is simulated and verified using the IEEE 30-node system as an example with the objective of the lowest integrated operating cost of the combined system, and the following conclusions are drawn:

• (1) Through the joint provision of system rotating standby by the photovoltaic power plant and the thermal power unit, the cost of system rotating standby is significantly reduced through the standby capacity optimisation method proposed in the paper while ensuring the reliable operation of the system, and the cost of system rotating standby capacity is reduced by 18.9% compared with that of the traditional system standby capacity provided by the thermal power unit alone.

• (2) The introduction of electric heating and CSP generation effectively relieves the regulation pressure of thermal power units and improves the system's level of wind and light consumption; the generation cost of thermal power units is reduced by 10.6%, the amount of wind and light abandoned is reduced by 87.6%, and the comprehensive operation cost of the system is reduced by 8.89%.

• (3) The process of system optimal scheduling is actually a mutual game of economy and reliability, and the scheduler needs to select the appropriate confidence level according to the actual operational needs to achieve a balance between the two.